# New constructions for the $n$-queens problem

## TL;DR

This paper explores new graph constructions for the $n$-queens problem using queen labelings of digraphs and the $igotimes_h$-product, providing novel methods to generate solutions and extending previous work.

## Contribution

It introduces a new construction method for the $n$-queens problem based on the $igotimes_h$-product and queen labelings of 1-regular digraphs, complementing prior results.

## Key findings

- Established a bijection between queen labelings and $n$-queens solutions.
- Developed a new construction method using the $igotimes_h$-product.
- Extended previous results by Pólya on $n$-queens problem solutions.

## Abstract

Let $D$ be a digraph, possibly with loops. A queen labeling of $D$ is a bijective function $l:V(G)\longrightarrow \{1,2,\ldots,|V(G)|\}$ such that, for every pair of arcs in $E(D)$, namely $(u,v)$ and $(u',v')$ we have (i) $l(u)+l(v)\neq l(u')+l(v')$ and (ii) $l(v)-l(u)\neq l(v')-l(u')$. Similarly, if the two conditions are satisfied modulo $n=|V(G)|$, we define a modular queen labeling. There is a bijection between (modular) queen labelings of $1$-regular digraphs and the solutions of the (modular) $n$-queens problem. The $\otimes_h$-product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the $\otimes_h$-product and some particular families of graphs. In this paper, we study some families of $1$-regular digraphs that admit (modular) queen labelings and present a new construction concerning to the (modular) $n$-queens problem in terms of the $\otimes_h$-product, which in some sense complements a previous result due to P\'olya.